Questions — Edexcel D2 (237 questions)

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Edexcel D2 Q2
8 marks Moderate -0.8
A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).
IIIIIIIV
I\(-4\)\(-5\)\(-2\)4
II\(-1\)1\(-1\)2
III05\(-2\)\(-4\)
IV\(-1\)3\(-1\)1
  1. Determine the play-safe strategy for each player. [4]
  2. Verify that there is a stable solution and determine the saddle points. [3]
  3. State the value of the game to \(B\). [1]
Edexcel D2 Q3
10 marks Standard +0.3
\includegraphics{figure_2} The network in Fig. 2 shows possible routes that an aircraft can take from \(S\) to \(T\). The numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. The airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. This is the minimax route.
  1. Complete the table in the answer booklet. [8]
  2. Hence obtain the minimax route from \(S\) to \(T\) and state the maximum amount of fuel used on any part of this route. [2]
Edexcel D2 Q4
8 marks Moderate -0.3
Andrew (\(A\)) and Barbara (\(B\)) play a zero-sum game. This game is represented by the following pay-off matrix for Andrew. $$A \begin{pmatrix} 3 & 5 & 4 \\ 1 & 4 & 2 \\ 6 & 3 & 7 \end{pmatrix}$$
  1. Explain why this matrix may be reduced to $$\begin{pmatrix} 3 & 5 \\ 6 & 3 \end{pmatrix}$$ [8]
  2. Hence find the best strategy for each player and the value of the game.
Edexcel D2 Q5
11 marks Moderate -0.5
An engineering company has 4 machines available and 4 jobs to be completed. Each machine is to be assigned to one job. The time, in hours, required by each machine to complete each job is shown in the table below.
Job 1Job 2Job 3Job 4
Machine 114587
Machine 221265
Machine 37839
Machine 424610
Use the Hungarian algorithm, \emph{reducing rows first}, to obtain the allocation of machines to jobs which minimises the total time required. State this minimum time. [11]
Edexcel D2 Q6
14 marks Moderate -0.3
The table below shows the distances, in km, between six towns \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\).
ABCDEF
A\(-\)85110175108100
B85\(-\)3817516093
C11038\(-\)14815673
D175175148\(-\)11084
E108160156110\(-\)92
F10093738492\(-\)
  1. Starting from \(A\), use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected. [4]
    1. Using your answer to part (a) obtain an initial upper bound for the solution of the travelling salesman problem. [2]
    2. Use a short cut to reduce the upper bound to a value less than 680. [4]
  3. Starting by deleting \(F\), find a lower bound for the solution of the travelling salesman problem. [4]
Edexcel D2 Q7
14 marks Standard +0.3
A steel manufacturer has 3 factories \(F_1\), \(F_2\) and \(F_3\) which can produce 35, 25 and 15 kilotomnes of steel per year, respectively. Three businesses \(B_1\), \(B_2\) and \(B_3\) have annual requirements of 20, 25 and 30 kilotomnes respectively. The table below shows the cost \(C_{ij}\) in appropriate units, of transporting one kilotome of steel from factory \(F_i\) to business \(B_j\).
Business
\(B_1\)\(B_2\)\(B_3\)
\(F_1\)10411
Factory \(F_2\)1258
\(F_3\)967
The manufacturer wishes to transport the steel to the businesses at minimum total cost.
  1. Write down the transportation pattern obtained by using the North-West corner rule. [2]
  2. Calculate all of the improvement indices \(I_{ij}\) and hence show that this pattern is not optimal. [5]
  3. Use the stepping-stone method to obtain an improved solution. [3]
  4. Show that the transportation pattern obtained in part (c) is optimal and find its cost. [4]
Edexcel D2 Q8
14 marks Moderate -0.8
\includegraphics{figure_4} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.
  1. Write down the source vertices. [2]
\includegraphics{figure_5} Figure 5 shows a feasible flow through the same network.
  1. State the value of the feasible flow shown in Fig. 5. [1]
Taking the flow in Fig. 5 as your initial flow pattern,
  1. use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow. [6]
  2. Show the maximal flow on Diagram 2 and state its value. [3]
  3. Prove that your flow is maximal. [2]
Edexcel D2 Q9
17 marks Moderate -0.3
T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
ProcessingBlendingPackingProfit (£100)
Morning blend3134
Afternoon blend2345
Evening blend4233
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x\), \(y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
  1. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. [4]
An initial Simplex tableau for the above situation is
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)32410035
\(s\)13201020
\(t\)24300124
\(P\)\(-4\)\(-5\)\(-3\)0000
  1. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage. [11]
T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
  1. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available. [2]
Edexcel D2 Q10
6 marks Moderate -0.3
While solving a maximizing linear programming problem, the following tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)00\(1\frac{1}{3}\)10\(-\frac{1}{3}\)\(\frac{5}{3}\)
\(y\)01\(3\frac{1}{3}\)01\(-\frac{1}{3}\)\(\frac{1}{3}\)
\(x\)10\(-3\)0\(-1\)\(\frac{1}{3}\)1
\(P\)00101111
  1. Explain why this is an optimal tableau. [1]
  2. Write down the optimal solution of this problem, stating the value of every variable. [3]
  3. Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\). [2]
Edexcel D2 Q11
11 marks Standard +0.3
A company wishes to transport its products from 3 factories \(F_1\), \(F_2\) and \(F_3\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \includegraphics{figure_5}
  1. On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added. [2]
    1. State the maximum flow along \(SF_1ABR\) and \(SF_2CR\). [2]
    2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles.
    Taking your answer to part (b)(ii) as the initial flow pattern,
    1. use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow. [7]
    2. Prove that your final flow is maximal.
Edexcel D2 Q12
11 marks Standard +0.3
\includegraphics{figure_2} A company has 3 warehouses \(W_1\), \(W_2\) and \(W_3\). It needs to transport the goods stored there to 2 retail outlets \(R_1\) and \(R_2\). The capacities of the possible routes, in van loads per day, are shown in Fig. 2. Warehouses \(W_1\), \(W_2\) and \(W_3\) have 14, 12 and 14 van loads respectively available per day and retail outlets \(R_1\) and \(R_2\) can accept 6 and 25 van loads respectively per day.
  1. On Diagram 1 on the answer sheet add a supersource \(W\) and a supersink \(R\) and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added. [3]
  2. State the maximum flow along
    1. \(W_1W_1R_1R\),
    2. \(W_2CR_2R\).
    [2]
  3. Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from \(W\) to \(R\). Show your working on Diagram 2. List each flow-augmenting route you find together with its flow. [5]
  4. From your final flow pattern, determine the number of van loads passing through \(B\) each day. [1]
Edexcel D2 2004 June Q1
4 marks Easy -2.0
In game theory explain what is meant by
  1. zero-sum game, [2]
  2. saddle point. [2]
(Total 4 marks)
Edexcel D2 2004 June Q2
9 marks Moderate -0.8
In a quiz there are four individual rounds, Art, Literature, Music and Science. A team consists of four people, Donna, Hannah, Kerwin and Thomas. Each of four rounds must be answered by a different team member. The table shows the number of points that each team member is likely to get on each individual round.
ArtLiteratureMusicScience
Donna31243235
Hannah16101922
Kerwin19142021
Thomas18152123
Use the Hungarian algorithm, reducing rows first, to obtain an allocation which maximises the total points likely to be scored in the four rounds. You must make your method clear and show the table after each stage. [9] (Total 9 marks)
Edexcel D2 2004 June Q3
12 marks Moderate -0.3
The table shows the least distances, in km, between five towns, \(A\), \(B\), \(C\), \(D\) and \(E\).
\(A\)\(B\)\(C\)\(D\)\(E\)
\(A\)\(-\)15398124115
\(B\)153\(-\)74131149
\(C\)9874\(-\)82103
\(D\)12413182\(-\)134
\(E\)115149103134\(-\)
Nassim wishes to find an interval which contains the solution to the travelling salesman problem for this network.
  1. Making your method clear, find an initial upper bound starting at \(A\) and using
    1. the minimum spanning tree method, [4]
    2. the nearest neighbour algorithm. [3]
  2. By deleting \(E\), find a lower bound. [4]
  3. Using your answers to parts (a) and (b), state the smallest interval that Nassim could correctly write down. [1]
(Total 12 marks)
Edexcel D2 2004 June Q4
14 marks Standard +0.3
Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. \(\begin{pmatrix} -4 & -1 & 3 \\ 2 & 1 & -2 \end{pmatrix}\)
  1. Show that there is no stable solution. [3]
  2. Find the best strategy for Emma and the value of the game to her. [8]
  3. Write down the value of the game to Freddie and his pay-off matrix. [3]
(Total 14 marks)
Edexcel D2 2004 June Q5
18 marks Moderate -0.8
  1. Describe a practical problem that could be solved using the transportation algorithm. [2]
A problem is to be solved using the transportation problem. The costs are shown in the table. The supply is from \(A\), \(B\) and \(C\) and the demand is at \(d\) and \(e\).
\(d\)\(e\)Supply
\(A\)5345
\(B\)4635
\(C\)2440
Demand5060
  1. Explain why it is necessary to add a third demand \(f\). [1]
  2. Use the north-west corner rule to obtain a possible pattern of distribution and find its cost.
    \(d\)\(e\)\(f\)Supply
    \(A\)5345
    \(B\)4635
    \(C\)2440
    Demand5060
    [5]
  3. Calculate shadow costs and improvement indices for this pattern. [5]
  4. Use the stepping-stone method once to obtain an improved solution and its cost. [5]
(Total 16 marks)
Edexcel D2 2004 June Q6
18 marks Standard +0.3
Joan sells ice cream. She needs to decide which three shows to visit over a three-week period in the summer. She starts the three-week period at home and finishes at home. She will spend one week at each of the three shows she chooses travelling directly from one show to the next. Table 1 gives the week in which each show is held. Table 2 gives the expected profits from visiting each show. Table 3 gives the cost of travel between shows. Table 1
Week123
ShowsA, B, CD, EF, G, H
Table 2
Show\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Expected Profit (£)900800100015001300500700600
Table 3
Travel costs (£)\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Home7080150809070
\(A\)180150
\(B\)140120
\(C\)200210
\(D\)200160120
\(E\)170100110
It is decided to use dynamic programming to find a schedule that maximises the total expected profit, taking into account the travel costs.
  1. Define suitable stage, state and action variables. [3]
  2. Determine the schedule that maximises the total profit. Show your working in a table. [12]
  3. Advise Joan on the shows that she should visit and state her total expected profit. [3]
(Total 18 marks)
Edexcel D2 2004 June Q7
13 marks Moderate -0.5
\includegraphics{figure_1} Figure 1 shows a capacitated directed network. The number on each arc is its capacity. \includegraphics{figure_2} Figure 2 shows a feasible initial flow through the same network.
  1. Write down the values of the flow \(x\) and the flow \(y\). [2]
  2. Obtain the value of the initial flow through the network, and explain how you know it is not maximal. [2]
  3. Use this initial flow and the labelling procedure on Diagram 1 below to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. Diagram 1 \includegraphics{figure_3} [5]
  4. Show your maximal flow pattern on Diagram 2. Diagram 2 \includegraphics{figure_4} [2]
  5. Prove that your flow is maximal. [2]
(Total 13 marks)
Edexcel D2 2004 June Q8
6 marks Moderate -0.3
A three-variable linear programming problem in \(x\), \(y\) and \(z\) is to be solved. The objective is to maximise the profit P. The following tableau was obtained.
Basic variable\(x\)\(y\)\(Z\)\(r\)\(s\)\(t\)Value
\(s\)30201\(-\frac{2}{3}\)\(\frac{2}{3}\)
\(r\)40\(\frac{7}{2}\)108\(\frac{9}{2}\)
\(y\)5170037
P30200863
  1. State, giving your reason, whether this tableau represents the optimal solution. [1]
  2. State the values of every variable. [3]
  3. Calculate the profit made on each unit of \(y\). [2]
(Total 6 marks)
Edexcel D2 2004 June Q9
7 marks Standard +0.3
\includegraphics{figure_5} The diagram above shows a network of roads represented by arcs. The capacity of the road represented by that arc is shown on each arc. The numbers in circles represent a possible flow of 26 from \(B\) to \(L\). Three cuts \(C_1\), \(C_2\) and \(C_3\) are shown on The diagram above.
  1. Find the capacity of each of the three cuts. [3]
  2. Verify that the flow of 26 is maximal. [1]
The government aims to maximise the possible flow from \(B\) to \(L\) by using one of two options. Option 1: Build a new road from \(E\) to \(J\) with capacity 5. or Option 2: Build a new road from \(F\) to \(H\) with capacity 3.
  1. By considering both options, explain which one meets the government's aim [3]
Edexcel D2 2004 June Q10
18 marks Moderate -0.5
Flatland UK Ltd makes three types of carpet, the Lincoln, the Norfolk and the Suffolk. The carpets all require units of black, green and red wool. For each roll of carpet, the Lincoln requires 1 unit of black, 1 of green and 3 of red, the Norfolk requires 1 unit of black, 2 of green and 2 of red, and the Suffolk requires 2 units of black, 1 of green and 1 of red. There are up to 30 units of black, 40 units of green and 50 units of red available each day. Profits of £50, £80 and £60 are made on each roll of Lincoln, Norfolk and Suffolk respectively. Flatland UK Ltd wishes to maximise its profit. Let the number of rolls of the Lincoln, Norfolk and Suffolk made daily be \(x\), \(y\) and \(z\) respectively.
  1. Formulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. [4]
This problem is to be solved using the Simplex algorithm. The most negative number in the profit row is taken to indicate the pivot column at each stage.
  1. Stating your row operations, show that after one complete iteration the tableau becomes
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)\(\frac{1}{2}\)0\(1\frac{1}{2}\)1\(-\frac{1}{2}\)010
    \(y\)\(\frac{1}{2}\)1\(\frac{1}{2}\)0\(\frac{1}{2}\)020
    \(t\)2000\(-1\)110
    P\(-10\)0\(-20\)04001600
    [4]
You may not need to use all of the tableaux.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
\(r\)
\(s\)
\(t\)
P
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
  1. Explain the practical meaning of the value 10 in the top row. [2]
    1. Perform one further complete iteration of the Simplex algorithm.
      Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
      Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
    2. State whether your current answer to part (d)(i) is optimal. Give a reason for your answer.
    3. Interpret your current tableau, giving the value of each variable. [8]
(Total 18 marks)
Edexcel D2 2006 June Q1
4 marks Easy -1.8
  1. State Bellman's principle of optimality. [1]
  2. Explain what is meant by a minimax route. [1]
  3. Describe a practical problem that would require a minimax route as its solution. [2]
(Total 4 marks)
Edexcel D2 2006 June Q2
Moderate -0.8
Three workers, \(P\), \(Q\) and \(R\), are to be assigned to three tasks, 1, 2 and 3. Each worker is to be assigned to one task and each task must be assigned to one worker. The cost, in hundreds of pounds, of using each worker for each task is given in the table below. The cost is to be minimised.
Cost (in £100s)Task 1Task 2Task 3
Worker \(P\)873
Worker \(Q\)956
Worker \(R\)1044
Formulate the above situation as a linear programming problem, defining the decision variables and making the objective and constraints clear. (Total 7 marks)
Edexcel D2 2006 June Q3
11 marks Moderate -0.5
A college wants to offer five full-day activities with a different activity each day from Monday to Friday. The sports hall will be used for these activities. Each evening the caretaker will prepare the hall by putting away the equipment from the previous activity and setting up the hall for the activity next day. On Friday evening he will put away the equipment used that day and set up the hall for the following Monday. The 5 activities offered are Badminton (\(B\)), Cricket nets (\(C\)), Dancing (\(D\)), Football coaching (\(F\)) and Tennis (\(T\)). Each will be on the same day from week to week. The college decides to offer the activities in the order that minimises the total time the caretaker has to spend preparing the hall each week. The hall is initially set up for Badminton on Monday. The table below shows the time, in minutes, it will take the caretaker to put away the equipment from one activity and set up the hall for the next.
To
\cline{2-6} \multicolumn{1}{c|}{Time}\(B\)\(C\)\(D\)\(F\)\(T\)
\(B\)--10815064100
\(C\)108--5410460
From \(D\)15054--150102
\(F\)64104150--68
\(T\)1006010268--
  1. Explain why this problem is equivalent to the travelling salesman problem. [2]
  2. Find the total time taken by the caretaker each week using this ordering. A possible ordering of activities is
    MondayTuesdayWednesdayThursdayFriday
    \(B\)\(C\)\(D\)\(F\)\(T\)
    [2]
  3. Starting with Badminton on Monday, use a suitable algorithm to find an ordering that reduces the total time spent each week to less than 7 hours. [3]
  4. By deleting \(B\), use a suitable algorithm to find a lower bound for the time taken each week. Make your method clear. [4]
(Total 11 marks)
Edexcel D2 2006 June Q4
11 marks Standard +0.3
During the school holidays four building tasks, rebuilding a wall (\(W\)), repairing the roof (\(R\)), repainting the hall (\(H\)) and relaying the playground (\(P\)), need to be carried out at a Junior School. Four builders, \(A\), \(B\), \(C\) and \(D\) will be hired for these tasks. Each builder must be assigned to one task. Builder \(B\) is not able to rebuild the wall and therefore cannot be assigned to this task. The cost, in thousands of pounds, of using each builder for each task is given in the table below.
Cost\(H\)\(P\)\(R\)\(W\)
\(A\)35119
\(B\)378--
\(C\)25107
\(D\)8376
  1. Use the Hungarian algorithm, reducing rows first, to obtain an allocation that minimises the total cost. State the allocation and its total cost. You must make your method clear and show the table after each stage. [9]
  2. State, with a reason, whether this allocation is unique. [2]
(Total 11 marks)